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The negative binomial can be applied in any situation in which there is a series of independent trials, each of which can result in either of just two outcomes. The distribution applies to the number of trials that occur before the designated outcome occurs. For example, if you start flipping a fair coin repeatedly the negative binomial distribution gives the number of times you must flip the coin until you see 'heads'. There are also 'everyday' applications in inventory control and the insurance industry. Please see the link.
Use the continuity correction when using the normal distribution to approximate a binomial distribution to take into account the binomial is a discrete distribution and the normal distribution is continuous.
Binomial distribution is learned about in most statistic courses. You could use them in experiments when there are two possible outcomes and each experiment is independent.
how are emeralds use in everyday life
Unless your "everyday life" involves work in some area of engineering, you won't use matrices in your everyday life.